Subgroups — C₂₃×#34995 #19182621

Order 5888.

2214 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2852	C₂	1	4	yes	maximal, complemented
2	C₂₃×#2643	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2510	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2493	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2481	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2480	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2315	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2315	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#2308	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#2308	C₂	1	1	yes	maximal, characteristic
11	C₂×C₂₃×#386	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂₃×#385	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#126	C₂²	1	4	yes	complemented
17	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
18	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂²×C₄²×C₂₃	C₂²	1	2	yes	complemented
25	C₂₃×#401	C₂²	1	4	yes	complemented
26	C₂₃×#400	C₂²	1	4	yes	complemented
27	C₂₃×#400	C₂²	1	4	yes	complemented
28	C₂₃×#400	C₂²	1	4	yes	complemented
29	C₂₃×#397	C₂²	1	4	yes	complemented
30	C₂₃×#397	C₂²	1	4	yes	complemented
31	C₂₃×#397	C₂²	1	4	yes	complemented
32	C₂₃×#396	C₂²	1	4	yes	complemented
33	C₂₃×#388	C₂²	1	4	yes	complemented
34	C₂₃×#388	C₂²	1	4	yes	complemented
35	C₂₃×#388	C₂²	1	4	yes	complemented
36	C₂₃×#387	C₂²	1	4	yes	complemented
37	C₂₃×#387	C₂²	1	4	yes	complemented
38	C₂₃×#387	C₂²	1	4	yes	complemented
39	C₂₃×#386	C₂²	1	2	yes	—
40	C₂₃×#386	C₂²	1	4	yes	—
41	C₂₃×#386	C₂²	1	4	yes	—
42	C₂₃×#386	C₂²	1	2	yes	—
43	C₂₃×#386	C₂²	1	2	yes	—
44	C₂₃×#386	C₂²	1	2	yes	—
45	C₂₃×#386	C₂²	1	4	yes	—
46	C₂₃×#386	C₂²	1	4	yes	—
47	C₂₃×#385	C₂²	1	4	yes	—
48	C₂₃×#385	C₂²	1	4	yes	—
49	C₂₃×#385	C₂²	1	4	yes	complemented
50	C₂₃×#385	C₂²	1	2	yes	complemented
51	C₂₃×#385	C₂²	1	2	yes	complemented
52	C₂₃×#385	C₂²	1	4	yes	complemented
53	C₂₃×#385	C₂²	1	4	yes	complemented
54	C₂₃×#383	C₂²	1	4	yes	complemented
55	C₂₃×#381	C₂²	1	2	yes	—
56	C₂₃×#381	C₂²	1	2	yes	—
57	C₂₃×#380	C₂²	1	4	yes	—
58	C₂₃×#380	C₂²	1	2	yes	complemented
59	C₂₃×#380	C₂²	1	2	yes	complemented
60	C₂₃×#379	C₂²	1	2	yes	—
61	C₂₃×#379	C₂²	1	2	yes	—
62	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
63	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
64	C₂×C₂₃×#95	C₂²	1	2	yes	—
65	C₂×C₂₃×#95	C₂²	1	2	yes	complemented
66	C₂×C₂₃×#95	C₂²	1	2	yes	complemented
67	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
68	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₂₃	—	2	4	no	—
70	C₂²×D₄×C₂₃	—	2	4	no	—
71	C₂²×D₄×C₂₃	—	2	4	no	—
72	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
73	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
74	C₂³×C₄×C₂₃	—	2	4	no	—
75	C₂³×C₄×C₂₃	—	4	4	no	—
76	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
79	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
80	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
82	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
84	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
85	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
86	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
87	C₂₃×#126	—	4	8	no	—
88	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
96	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
97	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
98	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
99	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
100	C₂×C₂²⋊C₄×C₂₃	—	4	4	no	—
101	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
103	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
106	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
110	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
117	C₂×C₄²×C₂₃	—	4	4	no	—
118	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
119	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
120	C₂₃×#95	C₂³	1	4	yes	—
121	C₂₃×#95	C₂³	1	4	yes	—
122	C₂₃×#95	C₂³	1	4	yes	—
123	C₂₃×#95	C₂³	1	4	yes	—
124	C₂₃×#95	C₂³	1	4	yes	—
125	C₂₃×#95	C₂³	1	4	yes	—
126	C₂₃×#95	C₂³	1	4	yes	—
127	C₂₃×#95	C₂³	1	4	yes	—
128	C₂₃×#95	C₂³	1	4	yes	—
129	C₂₃×#95	C₂³	1	4	yes	—
130	C₂₃×#95	C₂³	1	4	yes	—
131	C₂₃×#95	C₂³	1	4	yes	—
132	C₂₃×#95	C₂³	1	4	yes	—
133	C₂₃×#95	C₂³	1	4	yes	—
134	C₂⁴×C₂₃	—	2	2	no	—
135	C₂⁴×C₂₃	—	2	2	no	—
136	C₂⁴×C₂₃	—	2	2	no	—
137	C₂⁴×C₂₃	—	4	2	no	—
138	C₂⁴×C₂₃	—	4	2	no	—
139	C₂⁴×C₂₃	—	4	2	no	—
140	C₂⁴×C₂₃	—	4	2	no	—
141	C₂⁴×C₂₃	#41	1	4	yes	—
142	C₂⁴×C₂₃	#41	1	4	yes	—
143	C₂⁴×C₂₃	#41	1	4	yes	—
144	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
145	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
146	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
147	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
148	C₂×D₄×C₂₃	—	2	4	no	—
149	C₂×D₄×C₂₃	—	2	4	no	—
150	C₂×D₄×C₂₃	—	2	4	no	—
151	C₂×D₄×C₂₃	—	2	4	no	—
152	C₂×D₄×C₂₃	—	2	4	no	—
153	C₂×D₄×C₂₃	—	2	4	no	—
154	C₂×D₄×C₂₃	—	4	16	no	—
155	C₂²×C₄×C₂₃	—	2	2	no	—
156	C₂²×C₄×C₂₃	—	2	4	no	—
157	C₂²×C₄×C₂₃	—	2	4	no	—
158	C₂²×C₄×C₂₃	—	2	4	no	—
159	C₂²×C₄×C₂₃	—	2	4	no	—
160	C₂²×C₄×C₂₃	—	2	4	no	—
161	C₂²×C₄×C₂₃	—	2	4	no	—
162	C₂²×C₄×C₂₃	—	2	2	no	—
163	C₂²×C₄×C₂₃	—	2	2	no	—
164	C₂²×C₄×C₂₃	—	2	2	no	—
165	C₂²×C₄×C₂₃	—	2	2	no	—
166	C₂²×C₄×C₂₃	—	2	4	no	—
167	C₂²×C₄×C₂₃	—	2	2	no	—
168	C₂²×C₄×C₂₃	—	2	2	no	—
169	C₂²×C₄×C₂₃	—	2	4	no	—
170	C₂²×C₄×C₂₃	—	2	4	no	—
171	C₂²×C₄×C₂₃	—	4	2	no	—
172	C₂²×C₄×C₂₃	—	4	2	no	—
173	C₂²×C₄×C₂₃	—	4	2	no	—
174	C₂²×C₄×C₂₃	—	4	2	no	—
175	C₂²×C₄×C₂₃	—	4	2	no	—
176	C₂²×C₄×C₂₃	—	4	2	no	—
177	C₂²×C₄×C₂₃	—	4	2	no	—
178	C₂²×C₄×C₂₃	—	4	2	no	—
179	C₂²×C₄×C₂₃	—	4	2	no	—
180	C₂²×C₄×C₂₃	—	4	2	no	—
181	C₂²×C₄×C₂₃	—	4	2	no	—
182	C₂²×C₄×C₂₃	—	4	2	no	—
183	C₂²×C₄×C₂₃	—	4	2	no	—
184	C₂²×C₄×C₂₃	—	4	16	no	—
185	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
187	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
189	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
190	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
191	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
194	C₄⋊C₄×C₂₃	—	2	8	no	—
195	C₄⋊C₄×C₂₃	—	4	4	no	—
196	C₄⋊C₄×C₂₃	—	4	4	no	—
197	C₄⋊C₄×C₂₃	—	4	4	no	—
198	C₄⋊C₄×C₂₃	—	4	4	no	—
199	C₄⋊C₄×C₂₃	—	4	4	no	—
200	C₂²⋊C₄×C₂₃	—	2	4	no	—
201	C₂²⋊C₄×C₂₃	—	2	4	no	—
202	C₂²⋊C₄×C₂₃	—	2	4	no	—
203	C₂²⋊C₄×C₂₃	—	2	4	no	—
204	C₂²⋊C₄×C₂₃	—	2	4	no	—
205	C₂²⋊C₄×C₂₃	—	2	4	no	—
206	C₂²⋊C₄×C₂₃	—	2	4	no	—
207	C₂²⋊C₄×C₂₃	—	2	4	no	—
208	C₂²⋊C₄×C₂₃	—	2	4	no	—
209	C₂²⋊C₄×C₂₃	—	2	4	no	—
210	C₂²⋊C₄×C₂₃	—	2	8	no	—
211	C₂²⋊C₄×C₂₃	—	4	4	no	—
212	C₂²⋊C₄×C₂₃	—	4	4	no	—
213	C₂²⋊C₄×C₂₃	—	4	4	no	—
214	C₂²⋊C₄×C₂₃	—	4	4	no	—
215	C₂²⋊C₄×C₂₃	—	4	4	no	—
216	C₄²×C₂₃	—	2	8	no	—
217	C₄²×C₂₃	—	4	4	no	—
218	#34995	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂³×C₂₃	—	2	2	no	—
220	C₂³×C₂₃	—	2	2	no	—
221	C₂³×C₂₃	—	2	2	no	—
222	C₂³×C₂₃	—	2	2	no	—
223	C₂³×C₂₃	—	2	2	no	—
224	C₂³×C₂₃	—	2	2	no	—
225	C₂³×C₂₃	—	2	2	no	—
226	C₂³×C₂₃	—	2	2	no	—
227	C₂³×C₂₃	—	2	2	no	—
228	C₂³×C₂₃	—	2	4	no	—
229	C₂³×C₂₃	—	2	4	no	—
230	C₂³×C₂₃	—	4	4	no	—
231	C₂³×C₂₃	—	4	8	no	—
232	C₂³×C₂₃	—	4	4	no	—
233	C₂³×C₂₃	—	4	8	no	—
234	C₂³×C₂₃	—	4	8	no	—
235	C₂³×C₂₃	—	4	4	no	—
236	C₂³×C₂₃	—	8	2	no	—
237	C₂³×C₂₃	—	8	2	no	—
238	C₂³×C₂₃	—	8	4	no	—
239	C₂³×C₂₃	#126	1	16	yes	complemented
240	C₂³×C₂₃	C₂×#41	1	2	yes	—
241	C₂³×C₂₃	C₂×#41	1	2	yes	—
242	C₂³×C₂₃	C₂×#41	1	2	yes	—
243	C₂³×C₂₃	C₂×#41	1	2	yes	—
244	C₂³×C₂₃	C₂×#41	1	2	yes	—
245	C₂³×C₂₃	C₂×#41	1	2	yes	—
246	C₂³×C₂₃	C₂×#41	1	2	yes	—
247	C₂³×C₂₃	C₂×#41	1	2	yes	—
248	C₂³×C₂₃	C₂×#41	1	2	yes	—
249	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
250	D₄×C₂₃	—	4	8	no	—
251	D₄×C₂₃	—	4	8	no	—
252	C₂×C₄×C₂₃	—	2	8	no	—
253	C₂×C₄×C₂₃	—	2	2	no	—
254	C₂×C₄×C₂₃	—	2	2	no	—
255	C₂×C₄×C₂₃	—	2	4	no	—
256	C₂×C₄×C₂₃	—	2	4	no	—
257	C₂×C₄×C₂₃	—	2	2	no	—
258	C₂×C₄×C₂₃	—	2	2	no	—
259	C₂×C₄×C₂₃	—	2	4	no	—
260	C₂×C₄×C₂₃	—	2	4	no	—
261	C₂×C₄×C₂₃	—	2	2	no	—
262	C₂×C₄×C₂₃	—	2	2	no	—
263	C₂×C₄×C₂₃	—	2	2	no	—
264	C₂×C₄×C₂₃	—	2	2	no	—
265	C₂×C₄×C₂₃	—	2	4	no	—
266	C₂×C₄×C₂₃	—	2	4	no	—
267	C₂×C₄×C₂₃	—	2	4	no	—
268	C₂×C₄×C₂₃	—	2	2	no	—
269	C₂×C₄×C₂₃	—	2	4	no	—
270	C₂×C₄×C₂₃	—	2	4	no	—
271	C₂×C₄×C₂₃	—	2	2	no	—
272	C₂×C₄×C₂₃	—	2	4	no	—
273	C₂×C₄×C₂₃	—	2	4	no	—
274	C₂×C₄×C₂₃	—	2	4	no	—
275	C₂×C₄×C₂₃	—	2	4	no	—
276	C₂×C₄×C₂₃	—	2	4	no	—
277	C₂×C₄×C₂₃	—	4	8	no	—
278	C₂×C₄×C₂₃	—	4	8	no	—
279	C₂×C₄×C₂₃	—	4	8	no	—
280	C₂×C₄×C₂₃	—	4	8	no	—
281	C₂×C₄×C₂₃	—	4	8	no	—
282	C₂×C₄×C₂₃	—	4	4	no	—
283	C₂×C₄×C₂₃	—	4	4	no	—
284	C₂×C₄×C₂₃	—	8	2	no	—
285	C₂×C₄×C₂₃	—	8	2	no	—
286	C₂×C₄×C₂₃	—	8	4	no	—
287	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
288	#2852	C₂×C₂₃	1	4	yes	complemented
289	#2643	C₂×C₂₃	1	4	yes	complemented
290	#2510	C₂×C₂₃	1	4	yes	complemented
291	#2493	C₂×C₂₃	1	4	yes	complemented
292	#2481	C₂×C₂₃	1	4	yes	complemented
293	#2480	C₂×C₂₃	1	4	yes	complemented
294	#2315	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#2315	C₂×C₂₃	1	1	yes	characteristic, complemented
296	#2308	C₂×C₂₃	1	1	yes	characteristic
297	#2308	C₂×C₂₃	1	1	yes	characteristic
298	C₂×#386	C₂×C₂₃	1	1	yes	characteristic
299	C₂×#385	C₂×C₂₃	1	1	yes	characteristic, complemented
300	C₂×C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
301	C₂²×C₂₃	—	2	4	no	—
302	C₂²×C₂₃	—	2	4	no	—
303	C₂²×C₂₃	—	2	4	no	—
304	C₂²×C₂₃	—	2	2	no	—
305	C₂²×C₂₃	—	2	2	no	—
306	C₂²×C₂₃	—	2	2	no	—
307	C₂²×C₂₃	—	2	2	no	—
308	C₂²×C₂₃	—	2	2	no	—
309	C₂²×C₂₃	—	2	2	no	—
310	C₂²×C₂₃	—	2	4	no	—
311	C₂²×C₂₃	—	2	4	no	—
312	C₂²×C₂₃	—	2	4	no	—
313	C₂²×C₂₃	—	4	16	no	—
314	C₂²×C₂₃	—	4	2	no	—
315	C₂²×C₂₃	—	4	4	no	—
316	C₂²×C₂₃	—	4	2	no	—
317	C₂²×C₂₃	—	4	2	no	—
318	C₂²×C₂₃	—	4	4	no	—
319	C₂²×C₂₃	—	8	8	no	—
320	C₂²×C₂₃	C₂×#126	1	4	yes	—
321	C₂²×C₂₃	C₂×#126	1	4	yes	—
322	C₂²×C₂₃	C₂×#126	1	4	yes	—
323	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
324	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
325	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
326	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
327	C₂²×C₂₃	#585	1	2	yes	—
328	C₄×C₂₃	—	2	4	no	—
329	C₄×C₂₃	—	4	2	no	—
330	C₄×C₂₃	—	4	4	no	—
331	C₄×C₂₃	—	4	4	no	—
332	C₄×C₂₃	—	4	2	no	—
333	C₄×C₂₃	—	4	2	no	—
334	C₄×C₂₃	—	4	2	no	—
335	C₄×C₂₃	—	4	4	no	—
336	C₄×C₂₃	—	4	4	no	—
337	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
338	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
339	C₂×#126	C₂²×C₂₃	1	4	yes	complemented
340	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
341	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
342	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
343	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
344	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
345	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
346	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
347	C₂²×C₄²	C₂²×C₂₃	1	2	yes	complemented
348	#401	C₂²×C₂₃	1	4	yes	complemented
349	#400	C₂²×C₂₃	1	4	yes	complemented
350	#400	C₂²×C₂₃	1	4	yes	complemented
351	#400	C₂²×C₂₃	1	4	yes	complemented
352	#397	C₂²×C₂₃	1	4	yes	complemented
353	#397	C₂²×C₂₃	1	4	yes	complemented
354	#397	C₂²×C₂₃	1	4	yes	complemented
355	#396	C₂²×C₂₃	1	4	yes	complemented
356	#388	C₂²×C₂₃	1	4	yes	complemented
357	#388	C₂²×C₂₃	1	4	yes	complemented
358	#388	C₂²×C₂₃	1	4	yes	complemented
359	#387	C₂²×C₂₃	1	4	yes	complemented
360	#387	C₂²×C₂₃	1	4	yes	complemented
361	#387	C₂²×C₂₃	1	4	yes	complemented
362	#386	C₂²×C₂₃	1	4	yes	—
363	#386	C₂²×C₂₃	1	4	yes	—
364	#386	C₂²×C₂₃	1	2	yes	—
365	#386	C₂²×C₂₃	1	4	yes	—
366	#386	C₂²×C₂₃	1	2	yes	—
367	#386	C₂²×C₂₃	1	2	yes	—
368	#386	C₂²×C₂₃	1	2	yes	—
369	#386	C₂²×C₂₃	1	4	yes	—
370	#385	C₂²×C₂₃	1	4	yes	—
371	#385	C₂²×C₂₃	1	4	yes	—
372	#385	C₂²×C₂₃	1	4	yes	complemented
373	#385	C₂²×C₂₃	1	4	yes	complemented
374	#385	C₂²×C₂₃	1	4	yes	complemented
375	#385	C₂²×C₂₃	1	2	yes	complemented
376	#385	C₂²×C₂₃	1	2	yes	complemented
377	#383	C₂²×C₂₃	1	4	yes	complemented
378	#381	C₂²×C₂₃	1	2	yes	—
379	#381	C₂²×C₂₃	1	2	yes	—
380	#380	C₂²×C₂₃	1	4	yes	—
381	#380	C₂²×C₂₃	1	2	yes	complemented
382	#380	C₂²×C₂₃	1	2	yes	complemented
383	#379	C₂²×C₂₃	1	2	yes	—
384	#379	C₂²×C₂₃	1	2	yes	—
385	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
386	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	complemented
387	C₂×#95	C₂²×C₂₃	1	2	yes	—
388	C₂×#95	C₂²×C₂₃	1	2	yes	complemented
389	C₂×#95	C₂²×C₂₃	1	2	yes	complemented
390	C₂×C₂₃	—	2	4	no	—
391	C₂×C₂₃	—	2	4	no	—
392	C₂×C₂₃	—	2	4	no	—
393	C₂×C₂₃	—	8	2	no	—
394	C₂×C₂₃	C₂²×#126	1	1	yes	characteristic
395	C₂×C₂₃	#3459	1	2	yes	—
396	C₂×C₂₃	#3490	1	1	yes	characteristic
397	C₂×C₂₃	#3491	1	1	yes	characteristic
398	C₂×C₂₃	#3505	1	2	yes	—
399	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
400	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
401	C₂²×D₄	—	2	4	no	—
402	C₂²×D₄	—	2	4	no	—
403	C₂²×D₄	—	2	4	no	—
404	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
405	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
406	C₂³×C₄	—	2	4	no	—
407	C₂³×C₄	—	4	4	no	—
408	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
409	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
410	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
411	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
412	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
413	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
414	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
415	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
416	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
417	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
418	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
419	#126	—	4	8	no	—
420	C₂×C₄⋊C₄	—	4	4	no	—
421	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
422	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
423	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
424	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
425	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
426	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
427	C₂×C₂²⋊C₄	—	2	4	no	—
428	C₂×C₂²⋊C₄	—	2	4	no	—
429	C₂×C₂²⋊C₄	—	2	4	no	—
430	C₂×C₂²⋊C₄	—	2	4	no	—
431	C₂×C₂²⋊C₄	—	2	4	no	—
432	C₂×C₂²⋊C₄	—	4	4	no	—
433	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
435	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
436	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
437	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
438	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
439	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
440	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
441	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
442	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
443	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
444	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
445	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
446	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
447	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
448	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
449	C₂×C₄²	—	4	4	no	—
450	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
451	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
452	#95	C₂³×C₂₃	1	4	yes	—
453	#95	C₂³×C₂₃	1	4	yes	—
454	#95	C₂³×C₂₃	1	4	yes	—
455	#95	C₂³×C₂₃	1	4	yes	—
456	#95	C₂³×C₂₃	1	4	yes	—
457	#95	C₂³×C₂₃	1	4	yes	—
458	#95	C₂³×C₂₃	1	4	yes	—
459	#95	C₂³×C₂₃	1	4	yes	—
460	#95	C₂³×C₂₃	1	4	yes	—
461	#95	C₂³×C₂₃	1	4	yes	—
462	#95	C₂³×C₂₃	1	4	yes	—
463	#95	C₂³×C₂₃	1	4	yes	—
464	#95	C₂³×C₂₃	1	4	yes	—
465	#95	C₂³×C₂₃	1	4	yes	—
466	C₂₃	#34995	1	1	yes	characteristic, direct product factor, Sylow
467	C₂⁴	—	2	2	no	—
468	C₂⁴	—	2	2	no	—
469	C₂⁴	—	2	2	no	—
470	C₂⁴	—	4	2	no	—
471	C₂⁴	—	4	2	no	—
472	C₂⁴	—	4	2	no	—
473	C₂⁴	—	4	2	no	—
474	C₂⁴	#41×C₂₃	1	4	yes	—
475	C₂⁴	#41×C₂₃	1	4	yes	—
476	C₂⁴	#41×C₂₃	1	4	yes	—
477	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
478	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
479	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
480	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
481	C₂×D₄	—	2	4	no	—
482	C₂×D₄	—	2	4	no	—
483	C₂×D₄	—	2	4	no	—
484	C₂×D₄	—	2	4	no	—
485	C₂×D₄	—	2	4	no	—
486	C₂×D₄	—	2	4	no	—
487	C₂×D₄	—	4	16	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	4	no	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	—	2	4	no	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	2	4	no	—
494	C₂²×C₄	—	2	4	no	—
495	C₂²×C₄	—	2	2	no	—
496	C₂²×C₄	—	2	2	no	—
497	C₂²×C₄	—	2	2	no	—
498	C₂²×C₄	—	2	2	no	—
499	C₂²×C₄	—	2	4	no	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	2	no	—
502	C₂²×C₄	—	2	4	no	—
503	C₂²×C₄	—	2	4	no	—
504	C₂²×C₄	—	4	2	no	—
505	C₂²×C₄	—	4	2	no	—
506	C₂²×C₄	—	4	2	no	—
507	C₂²×C₄	—	4	2	no	—
508	C₂²×C₄	—	4	2	no	—
509	C₂²×C₄	—	4	2	no	—
510	C₂²×C₄	—	4	2	no	—
511	C₂²×C₄	—	4	2	no	—
512	C₂²×C₄	—	4	2	no	—
513	C₂²×C₄	—	4	2	no	—
514	C₂²×C₄	—	4	2	no	—
515	C₂²×C₄	—	4	2	no	—
516	C₂²×C₄	—	4	2	no	—
517	C₂²×C₄	—	4	16	no	—
518	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
519	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
520	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
521	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
522	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
523	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
524	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
525	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
526	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
527	C₄⋊C₄	—	2	8	no	—
528	C₄⋊C₄	—	4	4	no	—
529	C₄⋊C₄	—	4	4	no	—
530	C₄⋊C₄	—	4	4	no	—
531	C₄⋊C₄	—	4	4	no	—
532	C₄⋊C₄	—	4	4	no	—
533	C₂²⋊C₄	—	2	4	no	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₂²⋊C₄	—	2	4	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	4	no	—
540	C₂²⋊C₄	—	2	4	no	—
541	C₂²⋊C₄	—	2	4	no	—
542	C₂²⋊C₄	—	2	4	no	—
543	C₂²⋊C₄	—	2	8	no	—
544	C₂²⋊C₄	—	4	4	no	—
545	C₂²⋊C₄	—	4	4	no	—
546	C₂²⋊C₄	—	4	4	no	—
547	C₂²⋊C₄	—	4	4	no	—
548	C₂²⋊C₄	—	4	4	no	—
549	C₄²	—	2	8	no	—
550	C₄²	—	4	4	no	—
551	C₂³	—	2	2	no	—
552	C₂³	—	2	2	no	—
553	C₂³	—	2	2	no	—
554	C₂³	—	2	2	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	2	2	no	—
558	C₂³	—	2	2	no	—
559	C₂³	—	2	2	no	—
560	C₂³	—	2	4	no	—
561	C₂³	—	2	4	no	—
562	C₂³	—	4	4	no	—
563	C₂³	—	4	8	no	—
564	C₂³	—	4	4	no	—
565	C₂³	—	4	8	no	—
566	C₂³	—	4	8	no	—
567	C₂³	—	4	4	no	—
568	C₂³	—	8	2	no	—
569	C₂³	—	8	2	no	—
570	C₂³	—	8	4	no	—
571	C₂³	C₂₃×#126	1	16	yes	complemented
572	C₂³	C₂×#41×C₂₃	1	2	yes	—
573	C₂³	C₂×#41×C₂₃	1	2	yes	—
574	C₂³	C₂×#41×C₂₃	1	2	yes	—
575	C₂³	C₂×#41×C₂₃	1	2	yes	—
576	C₂³	C₂×#41×C₂₃	1	2	yes	—
577	C₂³	C₂×#41×C₂₃	1	2	yes	—
578	C₂³	C₂×#41×C₂₃	1	2	yes	—
579	C₂³	C₂×#41×C₂₃	1	2	yes	—
580	C₂³	C₂×#41×C₂₃	1	2	yes	—
581	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
582	D₄	—	4	8	no	—
583	D₄	—	4	8	no	—
584	C₂×C₄	—	2	8	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	4	no	—
588	C₂×C₄	—	2	4	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	4	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	2	4	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	4	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	4	no	—
605	C₂×C₄	—	2	4	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	4	no	—
609	C₂×C₄	—	4	8	no	—
610	C₂×C₄	—	4	8	no	—
611	C₂×C₄	—	4	4	no	—
612	C₂×C₄	—	4	8	no	—
613	C₂×C₄	—	4	8	no	—
614	C₂×C₄	—	4	8	no	—
615	C₂×C₄	—	4	4	no	—
616	C₂×C₄	—	8	2	no	—
617	C₂×C₄	—	8	2	no	—
618	C₂×C₄	—	8	4	no	—
619	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
620	C₂²	—	2	4	no	—
621	C₂²	—	2	4	no	—
622	C₂²	—	2	4	no	—
623	C₂²	—	2	2	no	—
624	C₂²	—	2	2	no	—
625	C₂²	—	2	2	no	—
626	C₂²	—	2	2	no	—
627	C₂²	—	2	2	no	—
628	C₂²	—	2	2	no	—
629	C₂²	—	2	4	no	—
630	C₂²	—	2	4	no	—
631	C₂²	—	2	4	no	—
632	C₂²	—	4	16	no	—
633	C₂²	—	4	2	no	—
634	C₂²	—	4	4	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	4	no	—
638	C₂²	—	8	8	no	—
639	C₂²	C₂×C₂₃×#126	1	4	yes	—
640	C₂²	C₂×C₂₃×#126	1	4	yes	—
641	C₂²	C₂×C₂₃×#126	1	4	yes	—
642	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
643	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
644	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
645	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
646	C₂²	C₂₃×#585	1	2	yes	—
647	C₄	—	2	4	no	—
648	C₄	—	4	2	no	—
649	C₄	—	4	4	no	—
650	C₄	—	4	4	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	2	no	—
653	C₄	—	4	2	no	—
654	C₄	—	4	4	no	—
655	C₄	—	4	4	no	—
656	C₂	—	2	4	no	—
657	C₂	—	2	4	no	—
658	C₂	—	2	4	no	—
659	C₂	—	8	2	no	—
660	C₂	C₂²×C₂₃×#126	1	1	yes	characteristic
661	C₂	C₂₃×#3459	1	2	yes	—
662	C₂	C₂₃×#3490	1	1	yes	characteristic
663	C₂	C₂₃×#3491	1	1	yes	characteristic
664	C₂	C₂₃×#3505	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂³×D₄×C₂₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂₃×#2852	C₂	semidirect
2. C₂₃×#2643	C₂	semidirect
3. C₂₃×#2510	C₂	semidirect
4. C₂₃×#2493	C₂	semidirect
5. C₂₃×#2481	C₂	semidirect
6. C₂₃×#2480	C₂	semidirect
7. C₂₃×#2315	C₂	semidirect
8. C₂₃×#2315	C₂	semidirect
12. C₂×C₂₃×#385	C₂	semidirect
13. C₂×C₄×C₂²⋊C₄×C₂₃	C₂	semidirect
16. C₂×C₂₃×#126	C₂²	semidirect
17. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
24. C₂²×C₄²×C₂₃	C₂²	semidirect
25. C₂₃×#401	C₂²	semidirect
26. C₂₃×#400	C₂²	semidirect
27. C₂₃×#400	C₂²	semidirect
28. C₂₃×#400	C₂²	semidirect
29. C₂₃×#397	C₂²	semidirect
30. C₂₃×#397	C₂²	semidirect
31. C₂₃×#397	C₂²	semidirect
32. C₂₃×#396	C₂²	semidirect
33. C₂₃×#388	C₂²	semidirect
34. C₂₃×#388	C₂²	semidirect
35. C₂₃×#388	C₂²	semidirect
36. C₂₃×#387	C₂²	semidirect
37. C₂₃×#387	C₂²	semidirect
38. C₂₃×#387	C₂²	semidirect
49. C₂₃×#385	C₂²	semidirect
50. C₂₃×#385	C₂²	semidirect
51. C₂₃×#385	C₂²	semidirect
52. C₂₃×#385	C₂²	semidirect
53. C₂₃×#385	C₂²	semidirect
54. C₂₃×#383	C₂²	semidirect
58. C₂₃×#380	C₂²	semidirect
59. C₂₃×#380	C₂²	semidirect
63. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
65. C₂×C₂₃×#95	C₂²	semidirect
66. C₂×C₂₃×#95	C₂²	semidirect
218. #34995	C₂₃	direct (N × H)
239. C₂³×C₂₃	#126	semidirect
288. #2852	C₂×C₂₃	semidirect
289. #2643	C₂×C₂₃	semidirect
290. #2510	C₂×C₂₃	semidirect
291. #2493	C₂×C₂₃	semidirect
292. #2481	C₂×C₂₃	semidirect
293. #2480	C₂×C₂₃	semidirect
294. #2315	C₂×C₂₃	semidirect
295. #2315	C₂×C₂₃	semidirect
299. C₂×#385	C₂×C₂₃	semidirect
300. C₂×C₄×C₂²⋊C₄	C₂×C₂₃	semidirect
339. C₂×#126	C₂²×C₂₃	semidirect
340. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
347. C₂²×C₄²	C₂²×C₂₃	semidirect
348. #401	C₂²×C₂₃	semidirect
349. #400	C₂²×C₂₃	semidirect
350. #400	C₂²×C₂₃	semidirect
351. #400	C₂²×C₂₃	semidirect
352. #397	C₂²×C₂₃	semidirect
353. #397	C₂²×C₂₃	semidirect
354. #397	C₂²×C₂₃	semidirect
355. #396	C₂²×C₂₃	semidirect
356. #388	C₂²×C₂₃	semidirect
357. #388	C₂²×C₂₃	semidirect
358. #388	C₂²×C₂₃	semidirect
359. #387	C₂²×C₂₃	semidirect
360. #387	C₂²×C₂₃	semidirect
361. #387	C₂²×C₂₃	semidirect
372. #385	C₂²×C₂₃	semidirect
373. #385	C₂²×C₂₃	semidirect
374. #385	C₂²×C₂₃	semidirect
375. #385	C₂²×C₂₃	semidirect
376. #385	C₂²×C₂₃	semidirect
377. #383	C₂²×C₂₃	semidirect
381. #380	C₂²×C₂₃	semidirect
382. #380	C₂²×C₂₃	semidirect
386. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
388. C₂×#95	C₂²×C₂₃	semidirect
389. C₂×#95	C₂²×C₂₃	semidirect
466. C₂₃	#34995	direct (N × H)
571. C₂³	C₂₃×#126	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#2852	2
C₂₃×#2643	2
C₂₃×#2510	2
C₂₃×#2493	2
C₂₃×#2481	2
C₂₃×#2480	2
C₂₃×#2315	2
C₂₃×#2308	2
C₂×C₂₃×#386	2
C₂×C₂₃×#385	2
C₂×C₄×C₂²⋊C₄×C₂₃	2
#34995	23

Subgroups — C23×#34995 #19182621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34995 #19182621